Why is there only one Iceland, and not a North-South chain of islands all down the Atlantic?

There is a ridge of elevated topography all along the Atlantic - the mid Atlantic ridge. This, like the bulk of the floor of the Atlantic was formed by magma produced at the mid Atlantic ridge. The reason why the mid Atlantic ridge is elevated, is because the the crust there is young and warm, making it relatively buoyant compared to the older crust further away from the ridge. The mid Atlantic ridge in the vicinity of Iceland is additionally elevated because it also has a mantle plume beneath it, which causes the crust above the plume head to bulge upward. The combination of both sitting above the plume, and being on the mid Atlantic ridge is sufficient to elevate the ocean floor above sea level.